Muckenhoupt-weighted $L_q(L_p)$ boundedness for time-space fractional nonlocal operators

We develop a weighted mixed-norm $L_q(L_p)$-estimates for solutions to fractional evolution equations of the form $${\mathrm{\partial }}_t^{\alpha }w(t,x)=\phi (\mathrm{\Delta })w(t,x)+h(t,x),w(0,\cdot )=w_0,t>0,x\in {\mathbb{R}}^d,$$ where ${\mathrm{\partial }}_t^{\alpha }$ denotes the Caputo derivative of $\alpha \in (0,1)$ and $\phi (\mathrm{\Delta })$ is a nonlocal operator associated with a Bernstein function $\phi$. For all $p,q\in (1,\mathrm{\infty })$ and $\gamma \in \mathbb{R}$, we prove the estimate \begin{align*} &\left| \partial_t^αw \right|{L_q(0,T,μ_2dt; H^{φ,γ}p(μ_1))} + \left| φ(Δ) w \right|{L_q(0,T,μ_2dt; H^{φ,γ}p(μ_1))} \ &\qquad\leq C \left( \left| h \right|{L_q(0,T,μ_2dt; H^{φ,γ}p(μ_1))} + \left| w_0 \right|{N{α,p,φ}} \right), \end{align*} where ${\mu }_1\in A_p({\mathbb{R}}^d)$ and ${\mu }_2\in A_q(\mathbb{R})$ are Muckenhoupt weights, and $N_{\alpha ,p,\phi }$ is a Banach space characterizing admissible initial data. In particular, when ${\mu }_2\equiv 1$ and $\alpha q>1$, $N_{\alpha ,p,\phi }$ coincides with the weighted Besov space $B_{p,q}^{\phi ,\gamma +2-\frac{2}{\alpha q}}({\mu }_1)$. The analysis employs tools from harmonic analysis, including the Fefferman–Stein inequality, Hardy-Littlewood maximal estimates in weighted mixed-norm spaces, and sharp function methods for bounding solution operators. These results extend and unify previous work by K.~H.~Kim et al, providing a general analytic framework for weighted $L_q(L_p)$-theory of time-space nonlocal evolution equations.